This resubmitted proposal for a phased-innovation award is to improve an algorithm for estimating non-rigid deformable motion, defined as motion where different parts of a biological specimen move at different rates or in different directions. The algorithm, developed by the PI (a computer scientist) and Co-PI (a biologist), is based on novel image processing techniques that combine tensor analysis with robust matching followed by motion interpolation to calculate dense velocity fields from a stack of (nine) successive images treated as a single image volume. Velocities are calculated at all pixels in the image along with an estimate of statistical confidence. The algorithm has been applied by the Co-PI to the deformable motion caused by the expansion of the plant root, and has revealed new insights and unexpected features of that process. The specific aims of the proposal are: 1) To generalize the algorithm by applying it to other examples, particularly to animal tissue culture cells moving in vitro and to neural crest cells migrating within the embryo. 2) To improve the algorithm computationally and develop adaptive routines with self-tuning parameters so that images having different textural features and motion quality can be handled. 3) To validate and evaluate the algorithm. And 4) To port the software to different platforms and provide a functional user interface, including flexibility to visualize the output graphically. Inspired by NIH-Image, we will make the software available for scientific evaluation (web-based downloads), and provide user driven improvements for biological deformable motion estimation. Software for estimating motion has focused on rigid motion, such as the movement of a car on a highway, or a bead along a microtubule, and algorithms for rigid motion are well known and available commercially. Quantifying the deformable motion typical of cells is difficult because the object changes as it moves. Current work on non-rigid motion is an active research area in computer science with applications in robot vision, computer graphics, and atmospheric science. However, development of non-rigid algorithms for images of relevance to biomedicine have been limited. Deformable motion is a hall-mark of biology, occurring in growth, embryogenesis, wound-healing, and metastasis, as well as in the movement of blood, organs, and whole organisms. Animal cell migration and guidance is the focus of considerable research on physiology and pathology. Quantifying motility algorithmically, at high accuracy, and at essentially every pixel of an image, should enjoy myriad applications throughout biomedicine.